12 Photometry The brightness of stars has, along with their locations, been studied by as- tronomers since ancient times. Prior to the 1860’s observers necessarily esti- mated brightness using their eyes expressing the result in the magnitude system that Ptolemy introduced in the second century. 12.1 A short history Modern instruments show that early measurements such as those made by Ptolemy and Tycho Brahe (who were both more interested in the position of ob- jects) have an internal precision of about 0.5m. Even a very skilled observer can do little better; al Sufi in the ninth century spent great effort on this problem and achieved a precision of some 0.4m. With a telescope several observers, such as the Herschels, were able to produce results of 0.1−0.3m by using comparisons with known sequences of standard brightness stars. Francois Arago suggested an optical/mechanical system by which adjusts the brightness of a comparison star until it matches the unknown star or dims the telescopic brightness of a star until it disappears. These systems are called visual photometers. Between 1879 and 1902 Harvard visual photometrists had measured the magnitudes of some 47 000 stars with a precision of about 0.08m and an accuracy of better than 0.25m. At this time in history (1900) the Pogson normal scale ∆m = −2.5 log(b1/b2) where b1 and b2 are the brightness of objectes 1 and 2 was standard amongst all astronomers. 1. The faintest stars visible to the naked eye, from the definition of the scale, are magnitude six. For point sources the brightness is increased by the use of a telescope by a factor G, called the light grasp. If the dark adapted human eye has a diameter of 7 mm, show that G ≈ 2 × 104d2 where d is the telscope diameter in meters. Show further that the limiting magnitude through a visually used telescope, mL is mL = 16.8+5log d. In the same period photography progressed and astronomers were able to record the light of stars too faint to be seen by eye in any telscope. An inter- national collaboration, the Carte du Ciel project, was started with the goal of photographing the entire sky and measuring the brightness of every star below 11.0m. However, photographic plates do not measure light linearly and it took several years, until 1900–1910, before a reliable photographic magnitude system 1 was established. The introduction of physical photometers in the period 1910– 1920 to objectively measure images on photographic plates eventually led to magnitudes being measured with uncertainties in the range 0.015 − 0.03m. Experiments with photoelectric work began in the early part of the 20th century and in the 1930’s with the introduction of vacuum-tube amplifiers de- tection limits on a 0.5 m telescope improved from 11.0m to 13.0m. Photomul- tiplier tubes, introduced during World War ii, improved the situation greatly and quickly became the instrument of choice for measuring brightness, with un- certainties of 0.005m in relative brightness. During the 1950 period to 1980 the RCA 1P21 photomultipliers were used by Harold Johnson to define the UBV system, which later was extended by the use of red sensitive photomultipliers into the infrared. At present CCDs and other modern solid-state detectors have mostly super- seded photomulitipliers. In the optical CCDs have superior efficiency, better stability, and a large multiplex advantage. From space, for example the Kepler mission for detecting occultations by extrasolar planets, achieves uncertainties below 10 µm over time scales of several weeks. 12.2 The response function A photometric device is sensitive over a restricted range of wavelengths called a bandpass. There are three types of bandpass photometry in astronomy. 12.2.1 Types of photometry Single-band photometry For applications such as finding planets via oc- cultations where one is only interested in measuring the fraction of light from the star blocked by a planet one needs only a single band. In this case one would generally want to construct a sequence of observations into a time series, i.e. a tabulation of brightness as a function of time, choosing a wide band to maximize signal and minimize the required exposure time and telescope size. An example of this is the Super Wasp telescope arrays placed on La Palma and in South Africa. Broadband multi-color photometry Broadband multi-color photometry measures a very low resolution spectrum by sampling the brightness in several different bands. Broad band in this sense means that the spectroscopic resolving power R = λc/∆λ< 10−15. These system attempt to choose bands that admit the maximum amount of light while still providing astrophysical information. The most typical example of such a system is the optical UBVRI system which uses bandwidths in the range 65−160 nm (R =4−7). This system can provide information on surface temperature, as well as (more limited) information on luminosity, metal content, and interstellar reddening for a wide variety of stars. Each band in a system such as this is is known as a color, so “two-color photometry” measures magnitudes in two separate bands. Usually one reports 2 the results of n-color photometric measurements by giving one magnitude and (n − 1) color indices. The magnitude tells the apparent brightness, and the indices tell about other astrophysical variables such as the surface temperature. Color can also have a second meaning: the difference between two magnitudes. For example, the results of two-color photometry in B and V will be reported as a V magnitude and one (B − V ) color. Narrow and intermediate-band photometry The intent of narrow band photometry (R > 50) is usually to isolate a specific line, molecular band, or other spectral feature. Common applications include the measurement of the strength of absorption features like Balmer-α or sodium D, or the ratio of the intensities of emission lines in gaseous nebulae. Intermediate-band photometry (15 <R< 50) measures spectroscopic features that cannot be resolved by broad bands but avoids the severe light loss of narrow-band photometry. Examples of such features include discontinuities in spectra, such as the Balmer discontinuity at 364.6 nm or very broad absorption features due to blended lines or molecular bands such as the band due to TiO in the spectra of M stars extending from 705 to 730 nm. 12.3 Magnitudes We can write the apparent magnitude of the source as ∞ mP = −2.5 log (FP)+ CP = −2.5log RP(λ)fλdλ + CP. Z0 Where mP is the bandpass magnitude, FP is the energy flux (irradiance) within the band, fλ is the monochromatic flux. The constant CP is chosen to conform to some standard scale (e.g. the magnitude of Vega is zero in the visual system). The function RP(λ) is called the response function of the entire observing system to the incident flux, it is the fraction of the energy of wavelength λ that will register on the photometer. Note that photometers count photons and therefore do not measure the energy directly. Thus we write the monochromatic photon flux λ φ(λ)= f hc λ and the quantity measured by by photon detectors is the photon flux within the band ∞ ∞ 1 Φ = R (λ)φ(λ)dλ = R f dλ P PP hc P λ Z0 Z0 where RPP is the photon response: the fraction of photons of wavelength λ detected by the system. It is also possible possible to define the monochromatic magnitude defined from the monochromatic flux: ′ hcφ(λ) ′ m = −2.5 log (f )+ C (λ)= −2.5log + C (λ) λ λ λ 3 C′(λ) is arbitrary, and is often chosen so that the monochromatic magnitude of Vega or som other standard is a constant at every wavelength. In which case C′(λ) is a strong function of λ. On the other hand C′(λ) can also be chosen as a constant function and at the monochromatic magnitude reflects the spectrum in energy units. 12.3.1 Response function implementation Both practical limits and intentional controls can determine the functional form of the response functions RP and RPP. The sensitivity of the detector limits the wavelength accessible. In some cases detector response alone sets the bandpass, in other cases the detector response defines only one edge of a given band. A filter is the usual method for intentionally delimiting the band by blocking all wavelengths except for those in a specific range. Filters can also serve as high-pass or low-pass elements to by only defining the lower or upper cutoff of a band. It is also possible to use a dispersing element to create a spectrum and then sampling discrete segments of the spectrum with one or more photometers. Such instruments are called spectrophotometers. These generally define bandpasses by using apertures, slots, or detectors of the proper size to select the desired segment of the spectrum. For ground based observations atmospheric transmission, Satm(λ), limits the wavelength that are accessible, and may define all or parts of a response function. 2. Find a plot of the Earth’s atmospheric transmission as a function of wave- length λ. Normally magnitudes are defined outside the Earth’s atmosphere, and as- tronomers must remove atmospheric effects during data reduction. 3. Give reasons as to why it is better to define the response function R(λ) outside the Earth’s atmosphere. What are the advantages and disadvan- tages of doing this? 12.3.2 Response function description There are a whole host of terms used to describe the response function R(λ).